System, method and apparatus for controlling converters using input-output linearization

ABSTRACT

The present invention provides a system, method and apparatus for controlling any type of converter using input-output linearization and leading-edge modulation. The controller includes a summing circuit connected to the converter to create a third voltage representing a difference between the first voltage and the second voltage. A gain circuit is connected to the summing circuit to adjust the third voltage by a proportional gain. A modulating circuit is connected to the gain circuit, the converter, the second voltage and the second current to create a control signal based on the second voltage, the adjusted third voltage, the first current and the second current. The control signal is used to control the converter. Typically, the first voltage is an output voltage from the converter, the second voltage is a reference voltage, the first current is an inductor current from the converter and the second current is a reference current.

FIELD OF THE INVENTION

The present invention relates generally to providing modulation signalsto electrical circuits and, more particularly, to a system, method andapparatus for controlling converters using input-output linearizationand leading-edge modulation.

BACKGROUND OF THE INVENTION

Power converters are used to convert one form of energy to another(e.g., AC to AC, AC to DC, DC to AC, and DC to DC) thereby making itusable to the end equipment, such as computers, automobiles,electronics, telecommunications, space systems and satellites, andmotors. Every application of power electronics involves some aspect ofcontrol. Converters are typically identified by their capability and/orconfigurations, such as, buck converters, boost converters, buck-boostconverters, boost-buck converters (Ĉuk), etc. For example, DC-DCconverters belong to a family of converters known as “switchingconverters” or “switching regulators.” This family of converters is themost efficient because the conversion elements switch from one state toanother, rather than needlessly dissipating power during the conversionprocess. Essentially there is a circuit with switches and twoconfigurations (each can be modeled as linear systems) in which theconverter resides according to the switch positions. The duty ratio (d)is the ratio indicating the time in which a chosen switch is in the “on”position while the other switch is in the “off” position, and this d isconsidered to be the control input. Input d is usually driven bypulse-width-modulation (PWM) techniques.

Switching from one state to another and the accompanying nonlinearity ofthe system causes problems. State space averaging reduces the switchingproblems to make the system, in general, a nonlinear averaged system.But, control of the system under these nonlinear effects becomesdifficult when certain performance objectives must be met. For the mostpart linearization is done through a Taylor series expansion. Nonlinearterms of higher orders are thrown away and a linear approximationreplaces the nonlinear system. This linearization method has proveneffective for stabilizing control loops at a specific operating point.However, use of this method requires making several assumptions, one ofthem being so-called “small signal operation.” This works well forasymptotic stability in the neighborhood of the operating point, butignores large signal effects which can result in nonlinear operation ofthe control loop when, for example, an amplifier saturates duringstartup, or during transient modes, such as load or input voltagechanges. Once nonlinear operation sets in, the control loop can haveequilibrium points unaccounted for in the linearization.

One of the most widely used methods of pulse-width modulation istrailing-edge modulation (TEM), wherein the on-time pulse begins on theclock and terminates in accordance with a control law. Unstable zerodynamics associated with TEM in the continuous conduction mode (CCM)prevent the use of an input-output feedback linearization because itwould result in an unstable operating point. The other control method isleading-edge modulation (LEM), wherein the on-time pulse begins inaccordance with a control law and terminates on the clock. Thedifference between LEM and TEM is that in TEM the pulse-width isdetermined by the instantaneous control voltage v_(c) prior to switchturn-off, whereas in LEM the pulse-width is determined by v_(c) prior toswitch turn-on.

There is, therefore, a need for a system, method and apparatus forcontrolling converters using input-output linearization that does notconstrain stability to one operating point, but rather to a set ofoperating points spanning the expected range of operation during startupand transient modes of operation.

SUMMARY OF THE INVENTION

The present invention provides a system, method and apparatus forcontrolling converters using input-output linearization that does notconstrain stability to one operating point, but rather to a set ofoperating points spanning the expected range of operation during startupand transient modes of operation. In particular, the present inventionuses leading edge modulation and input output linearization to computethe duty ratio of a boost converter or a buck-boost converter. Thepresent invention can also be applied to other converter types.Moreover, the parameters in this control system are programmable, andhence the algorithm can be easily implemented on a DSP or in silicon,such as an ASIC.

Notably, the present invention provides at least four advantagescompared to the dominant techniques currently in use for powerconverters. The combination of leading-edge modulation and input-outputlinearization provides a linear system instead of a nonlinear system. Inaddition, the “zero dynamics” becomes stable because the zeros are inthe left half plane instead of the right half plane. The presentinvention is also independent of stabilizing gain, as well as desiredoutput voltage or desired output trajectory.

More specifically, the present invention provides a system that includesa converter having a first voltage and a first current, a second voltagesource, a second current source, and a PWM modulator/controller. The PWMmodulator/controller includes a summing circuit connected to theconverter and the second voltage source to create a third voltagerepresenting a difference between the first voltage and the secondvoltage. A gain circuit is connected to the summing circuit to adjustthe third voltage by a proportional gain. A modulating circuit isconnected to the gain circuit, the converter, the second voltage and thesecond current source to create a control signal based on the secondvoltage, the adjusted third voltage, the first current and the secondcurrent. The control signal is then used to control the converter.

Note that the second voltage source and second current source can beintegrated into the PWM modulator/controller. In addition, the PWMmodulator/controller can be implemented using a digital signal processoror conventional electrical circuitry. Typically, the first voltage is anoutput voltage from the converter, the second voltage is a referencevoltage, the first current is an inductor current from the converter andthe second current is a reference current. The control signal of thepresent invention can be used to control a boost converter, a buck-boostconverter or other type of converter

In addition, the present invention provides a modulator/controller thatincludes a summing circuit, a gain circuit, a modulating circuit andvarious connections. The connections include a first connection toreceive a first voltage, a second connection to receive a secondvoltage, a third connection to receive a first current, a fourthconnection to receive a second current, and a fifth connection to outputa control signal. The summing circuit is connected to the firstconnection and the second connection to create a third voltagerepresenting a difference between the first voltage and the secondvoltage. The gain circuit is connected to the summing circuit to adjustthe third voltage by a proportional gain. The modulating circuit isconnected to the gain circuit, the second connection, the thirdconnection, the fourth connection and the fifth connection. Themodulation circuit creates a control signal based on the second voltage,the adjusted third voltage, the first current and the second current.Typically, the first voltage is an output voltage from the converter,the second voltage is a reference voltage, the first current is aninductor current from the converter and the second current is areference current. The control signal from the present invention can beused to control a boost converter, a buck-boost converter, or otherconverter type. Note also that the present invention can be applied toother non-linear systems.

The present invention also provides an apparatus that includes one ormore electrical circuits that provide a control signal to a boostconverter such that a duty cycle of the control signal is defined as$d = {- {\frac{\begin{matrix}{{\left\lbrack {{{CR}_{s}{R_{c}\left( {R + R_{c}} \right)}} + {L\quad\left( {R + R_{c}} \right)}} \right\rbrack\quad x_{1}} + \left\lbrack {{\left( {R + R_{\quad c}} \right)\quad R_{\quad c}C} + L} \right\rbrack} \\{{\frac{\left( {R + R_{c}} \right)}{R}y} - {\left( {R + R_{c}} \right)\quad R_{c}{Cu}_{0}} - {k\quad\left( {y - y_{o}} \right)}}\end{matrix}}{{LRx}_{1} + {\left\lbrack {\left( {R + R_{c}} \right)\quad R_{c}C} \right\rbrack\frac{\left( {R + R_{c}} \right)}{R}y}}.}}$Similarly, the present invention provides an apparatus that includes oneor more electrical circuits that provide a control signal to abuck-boost converter such that a duty cycle of the control signal isdefined as $d = {- {\frac{\begin{matrix}{{\left\lbrack {{{RR}_{\quad c}\quad C} + L} \right\rbrack\quad\frac{\left( {R + R_{\quad c}} \right)}{R}\quad y}\quad -} \\{{\begin{bmatrix}{{\left( {R + R_{\quad c}} \right)\quad R_{\quad c}R_{\quad s}\quad C} -} \\{L\quad\left( {R + R_{\quad c}} \right)}\end{bmatrix}\quad x_{\quad 1}} + {k\quad\left( {y - y_{0}} \right)}}\end{matrix}}{{\left( {R + R_{c}} \right)\quad R_{c}{Cy}} + {LRx}_{1} - {\left( {R + R_{c}} \right)\quad R_{c}{Cu}_{0}}}.}}$

Moreover, the present invention can be sold as a kit for engineers todesign and implement a PWM modulated converter. The kit may include adigital signal processor and a computer program embodied on a computerreadable medium for programming the digital signal processor to controlthe PWM modulated converter. The computer program may also include oneor more design tools. The digital signal processor includes a summingcircuit, a gain circuit, a modulating circuit and various connections.The various connections include a first connection to receive a firstvoltage, a second connection to receive a second voltage, a thirdconnection to receive a first current, a fourth connection to receive asecond current and a fifth connection to output a control signal. Thesumming circuit is connected to the first connection and the secondconnection to create a third voltage representing a difference betweenthe first voltage and the second voltage. The gain circuit is connectedto the summing circuit to adjust the third voltage by a proportionalgain. The modulating circuit is connected to the gain circuit, thesecond connection, the third connection, the fourth connection and thefifth connection. The modulation circuit creates a control signal basedon the second voltage, the adjusted third voltage, the first current andthe second current.

Furthermore, the present invention provides a method for controlling anon-linear system by receiving a first voltage, a second voltage, afirst current and a second current and creating a third voltagerepresenting a difference between the first voltage and the secondvoltage. The third voltage is then adjusted by a proportional gain. Thecontrol signal is created based on the second voltage, the adjustedthird voltage, the first current and the second current. The non-linearsystem is then controlled using the control signal. Typically, the firstvoltage is an output voltage from the converter, the second voltage is areference voltage, the first current is an inductor current from theconverter and the second current is a reference current. The controlsignal provides leading-edge modulation with input-output linearization,such that control of the non-linear system cannot be unstable. Moreover,the control signal is created using a first order system, and isindependent of a stabilizing gain, a desired output voltage or a desiredoutput trajectory. As a result, the present invention also provides aconverter controlled in accordance with the above-described method.Likewise, the present invention may include a computer program embodiedwithin a digital signal processor for controlling a non-linear systemwherein the steps of the above-described method are implemented as oneor more code segments.

Other features and advantages of the present invention will be apparentto those of ordinary skill in the art upon reference to the followingdetailed description taken in conjunction with the accompanyingdrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and further advantages of the invention may be betterunderstood by referring to the following description in conjunction withthe accompanying drawings, in which:

FIG. 1 is a block diagram of a system in accordance with the presentinvention;

FIG. 2 is a block diagram of a modulator/controller in accordance withthe present invention;

FIG. 3 is a flow chart of a control method in accordance with thepresent invention;

FIGS. 4A and 4B are graphs of trailing-edge modulation of a PWM signaland leading-edge modulation of a PWM signal;

FIG. 5 is circuit diagram of a boost converter and amodulator/controller in accordance with the present invention;

FIGS. 6A and 6B are linear circuit diagrams of a boost converter duringtime DTs and D'Ts, respectively in accordance with the presentinvention;

FIG. 7 is a graph of typical waveforms for the boost converter for thetwo switched intervals DTs and D'Ts in accordance with the presentinvention;

FIG. 8 is circuit diagram of a buck-boost converter and amodulator/controller in accordance with the present invention;

FIGS. 9A and 9B are linear circuit diagrams of a buck-boost converterduring time DTs and D'Ts, respectively in accordance with the presentinvention;

FIG. 10 is a graph of typical waveforms for the buck-boost converter forthe two switched intervals DTs and D'Ts in accordance with the presentinvention;

FIG. 11A is a graph of the boost converter output voltage, y, during y₀step from 28V to 32V to 28V in accordance with the present invention;

FIG. 11B is a graph of a close up view of the boost converter outputvoltage transition in FIG. 11A from 28V to 32V in accordance with thepresent invention;

FIG. 12 is a graph of the boost converter output voltage, y, during astep load change from 14 Ω to 7 Ω to 14 Ω in accordance with the presentinvention;

FIG. 13 is a graph of the buck-boost converter output voltage, y, duringy₀ step from −18V to −32V in accordance with the present invention;

FIG. 14 is a graph of the buck-boost converter output voltage, y, duringa step load change from 6 Ω to 3 Ω to 6 Ω in accordance with the presentinvention;

FIG. 15 is a schematic diagram of a buck-boost converter circuit for y₀step in accordance with the present invention;

FIG. 16 is an example of a schematic diagram of a control circuit andPWM modulator in accordance with the present invention;

FIG. 17 contains an example of schematic representations of the basicelements for calculation of duty ratio d in accordance with the presentinvention;

FIG. 18 is a graph of the buck-boost converter output voltage, y, duringy₀ step from −18V to −32V in accordance with the present invention;

FIG. 19 is a schematic diagram of a buck-boost converter circuit for Rstep in accordance with the present invention; and

FIG. 20 is graph of the buck-boost converter output voltage, y, during astep load change from 6 Ω to 3 Ω to 6 Ω in accordance with the presentinvention.

DETAILED DESCRIPTION OF THE INVENTION

While the making and using of various embodiments of the presentinvention are discussed in detail below, it should be appreciated thatthe present invention provides many applicable inventive concepts thatcan be embodied in a wide variety of specific contexts. The specificembodiments discussed herein are merely illustrative of specific ways tomake and use the invention and do not delimit the scope of theinvention.

The present invention provides a system, method and apparatus forcontrolling converters using input-output linearization that does notconstrain stability to one operating point, but rather to a set ofoperating points spanning the expected range of operation during startupand transient modes of operation. In particular, the present inventionuses leading-edge modulation and input-output linearization to computethe duty ratio of a boost converter or a buck-boost converter. Thepresent invention can also be applied to other converter types.Moreover, the parameters in this control system are programmable, andhence the algorithm can be easily implemented on a DSP or in silicon,such as an ASIC.

Notably, the present invention provides at least four advantagescompared to the dominant techniques currently in use for powerconverters. The combination of leading-edge modulation and input-outputlinearization provides a linear system instead of a nonlinear system. Inaddition, the “zero dynamics” becomes stable because the zeros are inthe left half plane instead of the right half plane. The presentinvention is also independent of stabilizing gain, as well as desiredoutput voltage or desired output trajectory.

As previously described, trailing-edge modulation for boost andbuck-boost converters operating in the continuous conduction mode givesrise to unstable zero dynamics where the linear part of the system aboutan operating point has a right half plane zero. In contrast, the presentinvention employs leading-edge modulation, along with some very simpledesign constraints, that change the zero dynamics so that the linearpart of the system has only left half plane zeros. Since the nonlinearsystem now possesses stable zero dynamics, input-output feedbacklinearization can be used. To apply this method, the actual output y ischosen as output function h(x), and y is repeatedly differentiated untilthe input u appears. The number of differentiations, r, is called therelative degree of the system. The present invention has a relativedegree r=1. The linearizing transformation for d is solved and used forthe control input. This transformation is local in nature, but it can beapplied in a neighborhood of any state space operating point in DC-DCconversion.

It is desirable to choose any operating point for the nonlinear system.This operating point can be made locally asymptotically stable by theabove process if a gain k is chosen to be positive. The gain k does nothave to be adjusted for each operating point, i.e., no gain schedulingis required. However, the reference input will have to be walked up,which is typical of soft-start operation, to insure convergence to theoperating point. Note that Proportional (P) and Proportional-Integral(PI) control loops can be added for robustness.

Now referring to FIG. 1, a block diagram of a system 100 in accordancewith the present invention is shown. The system includes a power source(voltage) 102 connected to a converter 104 that provides power to a load106. The converter 104 can be a boost converter, a buck-boost converteror other type of converter. The converter 104 is also connected to thePWM modulator/controller 108. The PWM modulator/controller 108 receiveshaving a first voltage 110, a second voltage 112, a first current 114and a second current 116. In most applications, the first voltage 110 isan output voltage from the converter 104, the second voltage 112 is areference voltage, the first current 114 is an inductor current from theconverter 104 and the second current 116 is a reference current. Notethat sources of the second voltage 112 and second current 116 can beintegrated within or external to the PWM modulator/controller 108. ThePWM modulator/controller 108 uses the first voltage 110, the secondvoltage 112, the first current 114 and the second current 116 togenerate a control signal 118 that is used to control the converter 104.The details of how the PWM modulator/controller 108 generates thecontrol signal 118 will be described in more detail below. In addition,the PWM modulator/controller 108 can be implemented using a digitalsignal processor or conventional electrical circuitry.

Referring now to FIG. 2, a block diagram of a modulator/controller 108in accordance with the present invention is shown. Themodulator/controller 108 includes a summing circuit 200, a gain circuit204, a modulating circuit 208 and various connections. The connectionsinclude a first connection to receive a first voltage 110, a secondconnection to receive a second voltage 112, a third connection toreceive a first current 114, a fourth connection to receive a secondcurrent 116, and a fifth connection to output a control signal 118. Thesumming circuit 200 is connected to the first connection and the secondconnection to create a third voltage (Δy) 202 representing a differencebetween the first voltage and the second voltage. The gain circuit 204is connected to the summing circuit 200 to adjust the third voltage (Δy)202 by a proportional gain (k). The modulating circuit 208 is connectedto the gain circuit 204, the second connection, the third connection,the fourth connection and the fifth connection. The modulation circuit208 creates a control signal 118 based on the second voltage, theadjusted third voltage (kΔy) 206, the first current and the secondcurrent. As implemented in the system of FIG. 1, the first voltage 110is an output voltage from the converter 104, the second voltage 112 is areference voltage, the first current 114 is an inductor current from theconverter 104 and the second current 116 is a reference current.Typically, the control signal 118 will be used to control a boostconverter, a buck-boost converter or other converter type. Note alsothat the present invention can be applied to other non-linear systems.

The present invention also provides an apparatus having one or moreelectrical circuits that provide a control signal 118 to a boostconverter such that a duty cycle of the control signal is defined as$d = {- {\frac{\begin{matrix}{{\left\lbrack {{{CR}_{s}{R_{c}\left( {R + R_{c}} \right)}} + {L\quad\left( {R + R_{c}} \right)}} \right\rbrack\quad x_{1}} + \left\lbrack {{\left( {R + R_{\quad c}} \right)\quad R_{\quad c}C} + L} \right\rbrack} \\{{\frac{\left( {R + R_{c}} \right)}{R}y} - {\left( {R + R_{c}} \right)\quad R_{c}{Cu}_{0}} - {k\quad\left( {y - y_{0}} \right)}}\end{matrix}}{{LRx}_{1} + {\left\lbrack {\left( {R + R_{c}} \right)\quad R_{c}C} \right\rbrack\frac{\left( {R + R_{c}} \right)}{R}y}}.}}$Similarly, the present invention provides an apparatus having one ormore electrical circuits that provide a control signal 118 to abuck-boost converter such that a duty cycle of the control signal isdefined as $d = {- {\frac{\begin{matrix}{{\left\lbrack {{{RR}_{\quad c}\quad C} + L} \right\rbrack\quad\frac{\left( {R + R_{\quad c}} \right)}{R}\quad y}\quad -} \\{{\begin{bmatrix}{{\left( {R + R_{\quad c}} \right)\quad R_{\quad c}R_{\quad s}\quad C} -} \\{L\quad\left( {R + R_{\quad c}} \right)}\end{bmatrix}\quad x_{\quad 1}} + \quad{k\quad\left( {y - y_{0}} \right)}}\end{matrix}}{{\left( {R + R_{c}} \right)\quad R_{c}{Cy}} + {LRx}_{1} - {\left( {R + R_{c}} \right)\quad R_{c}{Cu}_{0}}}.}}$

The present invention can be sold as a kit for engineers to design andimplement a PWM modulated converter. The kit may include a digitalsignal processor and a computer program embodied on a computer readablemedium for programming the digital signal processor to control the PWMmodulated converter. The computer program may also include one or moredesign tools. The digital signal processor includes a summing circuit200, a gain circuit 204, a modulating circuit 208 and variousconnections. The connections include a first connection to receive afirst voltage 110, a second connection to receive a second voltage 112,a third connection to receive a first current 114, a fourth connectionto receive a second current 116, and a fifth connection to output acontrol signal 118. The summing circuit 200 is connected to the firstconnection and the second connection to create a third voltage (Δy) 202representing a difference between the first voltage and the secondvoltage. The gain circuit 204 is connected to the summing circuit 200 toadjust the third voltage (Δy) 202 by a proportional gain (k). Themodulating circuit 208 is connected to the gain circuit 204, the secondconnection, the third connection, the fourth connection and the fifthconnection. The modulation circuit 208 creates a control signal 118based on the second voltage, the adjusted third voltage (kΔy) 206, thefirst current and the second current. As implemented in the system ofFIG. 1, the first voltage 110 is an output voltage from the converter104, the second voltage 112 is a reference voltage, the first current114 is an inductor current from the converter 104 and the second current116 is a reference current.

Now referring to FIG. 3, a flow chart 300 of a control method inaccordance with the present invention is shown. The present inventionprovides a method for controlling a non-linear system by receiving afirst voltage, a second voltage, a first current and a second current inblock 302, creating a third voltage representing a difference betweenthe first voltage and the second voltage in block 304 and adjusting thethird voltage by a proportional gain in block 306. The control signal iscreated in block 308 based on the second voltage, the adjusted thirdvoltage, the first current and the second current. The non-linear systemis controlled using the control signal in block 310. In one embodimentof the present invention, the first voltage is an output voltage fromthe converter, the second voltage is a reference voltage, the firstcurrent is an inductor current from the converter and the second currentis a reference current. Moreover, the nonlinear system can be a boostconverter, a buck-boost converter or any other converter type.

In one embodiment, the non-linear system is a boost converter and a dutycycle for the control signal is defined as $d = {- {\frac{\begin{matrix}{{\left\lbrack {{{CR}_{s}{R_{c}\left( {R + R_{c}} \right)}} + {L\quad\left( {R + R_{c}} \right)}} \right\rbrack\quad x_{1}} + \left\lbrack {{\left( {R + R_{\quad c}} \right)\quad R_{\quad c}C} + L} \right\rbrack} \\{{\frac{\left( {R + R_{c}} \right)}{R}y} - {\left( {R + R_{c}} \right)\quad R_{c}{Cu}_{0}} - {k\quad\left( {y - y_{0}} \right)}}\end{matrix}}{{LRx}_{1} + {\left\lbrack {\left( {R + R_{c}} \right)\quad R_{c}C} \right\rbrack\frac{\left( {R + R_{c}} \right)}{R}y}}.}}$

In yet another embodiment, the non-linear system is a buck-boostconverter and a duty cycle for the control signal is defined as$d = {- \frac{\begin{matrix}{{\left\lbrack {{{RR}_{\quad c}\quad C} + L} \right\rbrack\quad\frac{\left( {R + R_{\quad c}} \right)}{R}\quad y}\quad -} \\{{\begin{bmatrix}{{\left( {R + R_{\quad c}} \right)\quad R_{\quad c}R_{\quad s}\quad C} -} \\{L\quad\left( {R + R_{\quad c}} \right)}\end{bmatrix}\quad x_{\quad 1}} + \quad{k\quad\left( {y - y_{0}} \right)}}\end{matrix}}{{\left( {R + R_{c}} \right)\quad R_{c}{Cy}} + {LRx}_{1} - {\left( {R + R_{c}} \right)\quad R_{c}{Cu}_{0}}}}$

The control signal provides leading-edge modulation with input-outputlinearization, such that control of the non-linear system cannot beunstable. The control signal is created using a first order system, andis independent of a stabilizing gain, a desired output voltage or adesired output trajectory. As a result, the present invention alsoprovides a converter controlled in accordance with the above-describedmethod. Likewise, the present invention may include a computer programembodied within a digital signal processor for controlling a non-linearsystem wherein the steps of the above-described method are implementedas one or more code segments.

A more detailed description of the models used in the present inventionwill now be described. State space averaging allows the adding togetherof the contributions for each linear circuit during its respective timeinterval. This is done by using the duty ratio as a weighting factor oneach interval. As shown below, this weighting process leads to a singleset of equations for the states and the output. But first, the systemwill be described by its state space equations.

Assume that a linear system (A, b) is described by{dot over (x)}(t)=Ax(t)+bu(t)   (1)where A ε R^(n×n) is an ‘n×n’ matrix, and b ε R^(n) is an ‘n×1’ columnvector.

As previously mentioned, the duty ratio d is the ratio indicating thetime in which a chosen switch is in the “on” position while the otherswitch is in the “off” position. Ts is the switching period. The “on”time is then denoted as dTs. The general state equations for any type ofconverter consisting of two linear switched networks are as follows:

For 0≦t≦dTs,{dot over (x)}(t)=A _(α) x(t)+b _(α) u(t)   (2a)dTs≦t≦Ts,{dot over (x)}(t)=A _(β) x(t)+b _(β) u(t)   (2b)

The equations in (2a) can be combined with the equations in (2b) usingthe duty ratio, d, as a weighting factor. Thus,{dot over (x)}(t)=(dA _(α) +d'A _(β))x(t)+(db _(α) +d'b _(β))u(t)   (3)which can be written in the form of equation (1) as{dot over (x)}(t)=Ax(t)+bu(t)   (4)withA=dA _(α) +d′A _(β)andb=db _(α) +d'b _(β)where d'=1−d.

The buck cell is linear after state-space averaging and is therefore theeasiest topology to control. On the other hand, the boost and buck-boostcells are nonlinear and have non-minimum phase characteristics. Thesenonlinear cells will be described.

Beginning with a vector field f(x) and a scalar function h(x), the Liederivative of h with respect to f is denoted by L_(f)h. The derivativeis a scalar function and can be understood as the directional derivativeof h in the direction of the vector field f.

Definition: For a smooth scalar function h: R^(n)→R and a smooth vectorfield f: R^(n)→R^(n), the Lie derivative of h with respect to f isL_(f)h=Δhf.   (5)orL_(f)h=<Δh,f>  (6)where Δ represents the gradient and bold type represents a vector field,Δhf is matrix multiplication, and <Δh,f> is standard dot product onR^(n).

Lie derivatives of any order can be defined asL_(f) ⁰h=h   (7)L _(f) ^(i) h=Δ(L _(f) ^(i-1) h)f=L _(f) L _(f) ^(i-1) h.   (8)Also if g is another smooth vector field g: R^(n)→R^(n), thenL _(g) L _(f) h=Δ(L _(f) h)_(g).   (9)

Next, another operation is defined on vector fields, the Lie bracket.

Definition: For f and g as defined above, The Lie bracket of f and g isa vector field defined as[f,g]=Δg f−Δf g.   (10)where Δ is the Jacobian of the associated vector field.

The Lie bracket [f, g] is also denoted by adfg. This notation makes iteasier to define repeated brackets asad_(f) ⁰=g   (11)ad_(f) ^(i)g=[f,ad_(f) ^(i-1)g]  (12)

Definition: A linearly independent set of vector fields [f₁, f₂, . . . ,f_(m)] is said to be involutive if, and only if, there are scalarfunctions a_(ijk): R^(n)→R such that $\begin{matrix}{{{\left\lbrack {f_{i},f_{j}} \right\rbrack(x)} = {\sum\limits_{k = 1}^{m}\quad{{\alpha_{ijk}(x)}\quad{f_{k}(x)}\quad{\forall i}}}},j} & (13)\end{matrix}$Involutivity means that if one forms the Lie bracket of any pairs ofvector fields from the set [f₁, f₂, . . . , f_(m)] then the resultingvector field can be expressed as a linear combination of the originalset of vector fields.

Definition: A single input nonlinear system described by{dot over (x)}=f(x)+g(x)u   (14)where f(x), g(x) are C^(∞) vector fields on R^(n) and T is adiffeomorphism is input-state linearizable if∃Ω⊂R^(n) ∃z=T(x):Ω→R^(n) and ∃v=α(x)+β(x)u,β(x)≠0   (15)so that{dot over (z)}=Az+bv (16)is a controllable linear time invariant system on Ω in R^(n).Furthermore, to be input-state feedback linearizable, the following twoconditions must hold:

-   -   (i) the vector fields [g,[f,g], . . . ,[ad_(f) ^(n-1)g]] are        linearly independent in Ω; and    -   (ii) the distribution D=span{g,ad_(f)g, . . . ,ad_(f) ^(n-2)g}        is involutive in Ω.

The new state vector is nowz=[T ₁ L _(f) T ₁ . . . L _(f) ^(n-1) T ₁]^(T) =[T ₁ T ₂ . . . T_(n)]^(T)   (17)where the gradient of T₁ is orthogonal to the vector fields in D, and uis given byu=α(x)+β(x)v   (18)where $\begin{matrix}{\alpha = {- \frac{L_{f}^{n}T_{1}}{L_{g}L_{f}^{n - 1}T_{1}}}} & (19) \\{\beta = \frac{1}{L_{g}L_{f}^{n - 1}T_{1}}} & (20)\end{matrix}$Some variables above will change depending on the application. Thesewill be defined below.

Now, add an output y to the system. Unlike input-state linearizationwhere a transformation is first found to generate a new state vector anda new control input, here the output y is repeatedly differentiateduntil the input u appears, thereby showing a relationship between y andu.

For the nonlinear system $\begin{matrix}\begin{matrix}{\overset{.}{x} = {{f\quad(x)} + {g\quad(x)\quad u}}} \\{y = {h\quad(x)}}\end{matrix} & (21)\end{matrix}$and a point x₀, we differentiate y once to get{dot over (y)}=Δh{dot over (x)}=Δhf(x)+Δhg(x)u=L _(f) h(x)+L _(g) h(x)u.This is differentiated repeatedly until the coefficient of u isnon-zero. This procedure continues until for some integer r≦nL _(g) L _(f) ^(i) h(x)=0 for all x near x₀ and 0≦i≦r−2andL _(g) L _(f) ^(r−1) h(x ₀)≠0Then $\begin{matrix}{u = {- \frac{\left( {{L_{f}^{r}h\quad(x)} + v} \right)}{L_{g}L_{f}^{r - 1}h\quad(x)}}} & (22)\end{matrix}$and, for v=0, results in a multiple integrator system with transferfunction $\begin{matrix}{{H(s)} = \frac{1}{s^{r}}} & (23)\end{matrix}$State feedback can be added for pole placement withv=c ₀ h(x)+c ₁ L _(f) h(x)+c ₂ L _(f) ² h(x)+ . . . +c _(r−1) L _(f)^(r−1) h(x),where c₀,c₁, . . . ,c_(r−1) are constants to be chosen, and the integerr is the relative degree of the system (2-21). It is the number ofdifferentiations required before u appears.

The first r new coordinates are found as above by differentiating theoutput h(x) $\begin{matrix}\begin{matrix}{{\overset{.}{z}}_{1} = {{L_{f}{h\left( {x(t)} \right)}} = z_{2}}} \\{{\overset{.}{z}}_{2} = {{L_{f}^{2}{h\left( {x(t)} \right)}} = z_{3}}} \\\cdots \\{{\overset{.}{z}}_{r - 1} = {{L_{f}^{r - 1}{h\left( {x(t)} \right)}} = z_{r}}} \\{{\overset{.}{z}}_{r} = {{L_{f}^{r}{h\left( {x(t)} \right)}} + {L_{g}L_{f}^{r - 1}{h\left( {x(t)} \right)}{u(t)}}}}\end{matrix} & (24)\end{matrix}$Since x(t)=Φ⁻¹(z(t)), leta(z)=L _(g) L _(f) ^(r−1) h(Φ⁻¹(z))b(z)=L _(f) ^(r) h(Φ⁻¹(z))which is recognized from (22) that a(z) is the denominator term and b(z)is the numerator. Now{dot over (z)} _(r) =b(z(t))+a(z(t))u(t)where a(z(t)) is nonzero for all z in a neighborhood of z⁰.

To find the remaining n−r coordinates, let $\xi = \begin{bmatrix}z_{1} \\\vdots \\z_{r}\end{bmatrix}$ and $\eta = {\begin{bmatrix}z_{r + 1} \\\vdots \\z_{n}\end{bmatrix}.}$Here z_(r+1), . . . , z_(n) are added to z₁, . . . z_(r) to provide alegitimate coordinate system. With this notation we can write the newcoordinates in normal form as $\begin{matrix}\begin{matrix}{{\overset{.}{z}}_{1} = z_{2}} \\{{\overset{.}{z}}_{2} = z_{3}} \\\cdots \\{{\overset{.}{z}}_{r - 1} = z_{r}} \\{{\overset{.}{z}}_{r} = {{b\left( {\xi,\eta} \right)} + {{a\left( {\xi,\eta} \right)}u}}} \\{\overset{.}{\eta} = {{q\left( {\xi,\eta} \right)} + {{p\left( {\xi,\eta} \right)}u}}} \\{y = z_{1}}\end{matrix} & (25)\end{matrix}$

The equation for {dot over (η)} represents the n−r equations for whichno special form exists. The general equation, however, if the followingcondition holdsL _(g)Φ_(i)(x)=0is reduced{dot over (η)}=q(ξ,η)and the input u does not appear.

In general, the new nonlinear system is described by $\begin{matrix}{{\overset{.}{\xi} = {{A\quad\xi} + {B\quad\upsilon}}}{\overset{.}{\eta} = {{q\left( {\xi,\eta} \right)} + {{p\left( {\xi,\eta} \right)}u}}}{y = {C\quad\xi}}} & \left( {{26a},b,c} \right)\end{matrix}$with the matrices A, B, and C in normal form, andv=b(ξ, η)+a(ξ, η)u

If r=n, input-output linearization leads to input-state linearization.If r<n, then there are n−r equations describing the internal dynamics ofthe system. The zero dynamics, obtained by setting ξ=0 in equation (26b)and solving for η, are very important in determining the possiblestabilization of the system (21). If these zero dynamics are non-minimumphase then the input-output linearization in (22) cannot be used. If,however, the zero dynamics are minimum phase it means that poleplacement can be done on the linear part of (26a) using (22) and thesystem will be stable.

In the sequel, the bold letter used to indicate vector fields will onlybe used when the context is ambiguous as to what is meant. Otherwise,non-bold letters will be used. For a boost converter the driving voltageu(t), the current x_(i) through the inductor, and the voltage x₂ acrossthe capacitor are restricted to be positive, nonnegative, and positive,respectively. Only the continuous conduction mode (CCM) is considered.The duty ratio d is taken to be the control input and is constrained by0≦d≦1. The Cuk-Middlebrook averaged nonlinear state equations are usedto find a feedback transformation that maps these state equations to acontrollable linear system. This transformation is one-to-one with therestrictions on u(t), x₁ and x₂ just mentioned and with additionalrestrictions involving {dot over (u)}. These additional restrictions arenot needed if u(t) is a constant, as in DC-DC conversion. It isinteresting to note that restrictions on {dot over (u)} are unnecessaryfor the boost converter even if u(t) changes with time. The nonlinearsystem is said to be feedback linearizable or feedback linearized.Through the feedback transformation, the same second order linear systemfor every operating point can be seen.

The new switching model of the present invention will now be describedin more detail. The physical component parasitics R_(s), the DC seriesresistance of the filter inductor L, and R_(c), the equivalent seriesresistance of the filter capacitor C, now need to be included sinceR_(c) especially plays a central role in the analysis to follow.

The system in accordance with the present invention is of the form{dot over (x)}=f(x)+g(x)dy=h(x)   (27)With this in mind, the state equations are derived to include parasiticsR_(s) and R_(c).

There are four basic cells for fixed frequency PWM converters. They arethe buck, boost, buck-boost, and boost-buck (Ćuk) topologies. Manyderivations extend the basic cells in applications where isolation canbe added between input and output via transformers, however, theoperation can be understood through the basic cell. Each cell containstwo switches. Proper operation of the switches results in atwo-switch-state topology. In this regime, there is a controlling switchand a passive switch that are either on or off resulting in two “on”states. In contrast, a three state converter would consist of threeswitches, two controlling switches and one passive switch, resulting inthree “on” states.

The control philosophy used to control the switching sequence ispulse-width-modulation (PWM). A control voltage v_(c) is compared with aramp signal (“sawtooth”) and the output pulse width is the result ofv_(c)>v_(m). This is shown in FIG. 4A. A new cycle is initiated on thenegative slope of the ramp. The pulse ends when v_(c)<v_(m) which causesmodulation to occur on the trailing edge. This gives it the name“trailing-edge modulation.”

The difference between leading-edge modulation (LEM) and theconventionally used trailing-edge modulation (TEM) is that in TEM (FIG.4A) the pulse-width is determined by the instantaneous control voltagev_(c) prior to switch turn-off, whereas in LEM (FIG. 4B) the pulse-widthis determined by v_(c) prior to switch turn-on. The reason that samplingis “just prior” to switch commutation is that the intersection of v_(c)and v_(m) determines the new state of the switch. Notice that in FIG. 4Bthe sawtooth ramp has a negative slope.

Now referring to FIG. 5, a circuit diagram 500 of a boost converter anda modulator/controller 502 in accordance with the present invention isshown. The specifics of the boost converter are well known. In thiscase, S2 is implemented with a diode and S1 is implemented with anN-channel MOSFET. FIGS. 6A and 6B are linear circuit diagrams 600 and650 of the boost converter in FIG. 5 during time DTs and D'Ts,respectively. The converter 500 operates as follows: u₀ provides powerto the circuit during S1 conduction time (FIG. 6A) storing energy ininductor L. During this time S2 is biased off. When S1 turns off, theenergy in L causes the voltage across L to reverse polarity. Since oneend is connected to the input source, u₀, it remains clamped while theother end forward biases diode S2 and clamps to the output. Currentcontinues to flow through L during this time (FIG. 6B). When S1 turnsback on, the cycle repeats. FIG. 7 illustrates the typical waveforms forthe boost converter for the two switched intervals DTs and D'Ts.

The DC transfer function needs to be determined in order to know how theoutput, y, across the load R is related to the input u₀ at zerofrequency. In steady state, the volt-second integral across L is equalto zero. Thus,

∫₀ ^(T) ^(s) v_(L)dt=0   (28)

where Ts is the switching period. Therefore, the volt-seconds during theon-time must equal the volt-seconds during the off-time. Using thisvolt-second balance constraint one can derive an equation forvolt-seconds during the on-time of S1 (DTs) and another equation forvolt-seconds during the off-time of S1 (D'Ts).

The parasitics are eliminated by setting R_(s)=0 and R_(c)=0.

During time DTs:DT_(x)v_(L)=DT_(s)u₀   (29)During time D'Ts:D'T _(s) v _(L) =D'T _(s) x ₂ −D'T _(s) u ₀   (30)Since by equation (28)DT_(s)v_(L)=D'T_(s)v_(L)the RHS of equation (29) is set equal to the RHS of equation (30)resulting in $\begin{matrix}{\frac{x_{2}}{u_{0}} = \frac{1}{D^{\prime}}} & (31)\end{matrix}$Equation (31) is the ideal duty ratio equation for the boost cell. IfR_(s) and R_(c) are both non-zero then $\begin{matrix}{\frac{y}{u_{0} - {x_{1}R_{s}}} = \frac{1}{D^{\prime}}} & (32)\end{matrix}$

The output y is $\begin{matrix}{y = {{D^{\prime}\frac{{RR}_{c}}{R + R_{c}}x_{1}} + {\frac{R}{R + R_{c}}x_{2}}}} & (33)\end{matrix}$During dTs:${\overset{.}{x}}_{1} = {{\frac{1}{L}u} - {\frac{R_{s}}{L}x_{1}}}$${\overset{.}{x}}_{2} = {{- \frac{1}{C\left( {R + R_{c}} \right)}}x_{2}}$During (1−d)Ts:${\overset{.}{x}}_{1} = {\frac{1}{L}\left\lbrack {{{- \left( {R_{s} + \frac{{RR}_{c}}{R + R_{c}}} \right)}x_{1}} - {\frac{R}{R + R_{c}}x_{2}} + u_{0}} \right\rbrack}$${\overset{.}{x}}_{2} = {{\frac{R}{C\left( {R + R_{c}} \right)}x_{1}} - {\frac{1}{C\left( {R + R_{c}} \right)}x_{2}}}$Combining, the averaged equations are: $\begin{matrix}{{{\overset{.}{x}}_{1} = {{\frac{1}{L}u_{0}} - {\frac{R_{s}}{L}x_{1}} - {\frac{{RR}_{c}}{L\left( {R + R_{c}} \right)}{x_{1}\left( {1 - d} \right)}} - {\frac{R}{L\left( {R + R_{c}} \right)}{x_{2}\left( {1 - d} \right)}}}}{{\overset{.}{x}}_{2} = {{\frac{R}{C\left( {R + R_{c}} \right)}{x_{1}\left( {1 - d} \right)}} - {\frac{1}{C\left( {R + R_{c}} \right)}x_{2}}}}{y = {{\frac{R}{\left( {R + R_{c}} \right)}x_{2}} + {\frac{{RR}_{c}}{\left( {R + R_{c}} \right)}{x_{1}\left( {1 - d} \right)}}}}} & \left( {{34a},b,c} \right)\end{matrix}$where R_(s) is the dc resistance of L and R_(c) is the equivalent seriesresistance of C.

In standard form: $\begin{matrix}{{{\overset{.}{x}}_{1} = {\frac{u_{0}}{L} - {\frac{R}{L\left( {R + R_{c}} \right)}x_{2}} - {\left( {\frac{R_{s}}{L} + \frac{{RR}_{c}}{L\left( {R + R_{c}} \right)}} \right)x_{1}} + {\left( {{\frac{{RR}_{c}}{L\left( {R + R_{c}} \right)}x_{1}} + {\frac{R}{L\left( {R + R_{c}} \right)}x_{2}}} \right)d}}}{{\overset{.}{x}}_{2} = {{{- \frac{1}{C\left( {R + R_{c}} \right)}}x_{2}} + {\frac{R}{C\left( {R + R_{c}} \right)}x_{1}} - {\frac{R}{C\left( {R + R_{c}} \right)}x_{1}d}}}{y = {{\frac{{RR}_{c}}{\left( {R + R_{c}} \right)}x_{1}} + {\frac{R}{\left( {R + R_{c}} \right)}x_{2}}}}} & \left( {{35a},b,c} \right)\end{matrix}$

Here it is assumed that leading-edge modulation is used so that samplingof the output y only takes place during the interval (1−d)T_(s).Therefore, the weighting factor (1−d) in equation (34c) for y has beenremoved because when the sample is taken the data represents both termsas shown in equation (35c). In the present analysis the effects ofsampling (complex positive zero pair at one-half the sampling frequency)have been ignored.

The input-output linearization for the boost converter will now bediscussed. The output, y, only needs to be differentiated once beforethe control d appears. $\begin{matrix}{{y = {\frac{R}{R + R_{c}}\left( {x_{2} + {R_{c}x_{1}}} \right)}}{\overset{.}{y} = {\frac{R}{R + {Rc}}\left( {{\overset{.}{x}}_{2} + {R_{c}{\overset{.}{x}}_{1}}} \right)}}} & (36) \\{\overset{.}{y} = {\left( \frac{R}{R + {Rc}} \right)\left( {{\frac{1}{C\left( {R + R_{c}} \right)}\left( {{- x_{2}} + {Rx}_{1} - {{Rx}_{1}d}} \right)} + {\frac{R_{c}}{L}\left( {u_{0} - x_{2} - {\left( {R_{s} + \frac{{RR}_{c}}{R + R_{c}}} \right)x_{1}} + {\left( {{R_{c}x_{1}} + {\frac{R}{R + R_{c}}x_{2}}} \right)d}} \right)}} \right)}} & \left( {{37a},b} \right)\end{matrix}$

Substituting for x₂ from (36), setting {dot over (y)} to zero andsolving for d we get, $\begin{matrix}{d = {- \frac{\begin{matrix}{{\left\lbrack {{{CR}_{s}{R_{c}\left( {R + R_{c}} \right)}} + {L\left( {R + R_{c}} \right)}} \right\rbrack x_{1}} + \left\lbrack {{\left( {R + R_{c}} \right)R_{c}C} + L} \right\rbrack} \\{{\frac{\left( {R + R_{c}} \right)}{R}y} - {\left( {R + R_{c}} \right)R_{c}{Cu}_{0}} - {k\left( {y - y_{0}} \right)}}\end{matrix}}{{LRx}_{1} + {\left\lbrack {\left( {R + R_{c}} \right)R_{c}C} \right\rbrack\frac{\left( {R + R_{c}} \right)}{R}y}}}} & (38)\end{matrix}$where the error term k(y−y₀) has been added. Here y₀ is the desiredoutput corresponding to x₁₀ and x₂₀ through equation (36). The notationhas changed and k=c₀ in equation (22), and the control input is now dinstead of u. Here (x₁₀, x₂₀) is an equilibrium point of the boostconverter.

Implementation of this control differs from others that correct thestate before the transformation T, since the control is now part of thetransformation as shown in equation (38) where it is seen that k(y−y₀)is in the numerator, and k is the proportional gain. The controlimplementation is shown in FIG. 2.

Local linearization of the boost converter will now be discussed toobtain a transfer function. A Taylor Series linearization is used on thenonlinear system (35abc) to linearize about an operating point, x₁₀,x₂₀, D and obtain the transfer function. As before{circumflex over (x)} ₁ =x ₁ −x ₁₀ , {circumflex over (x)} ₂ =x ₂ −x ₂₀, ŷ=y−y ₀ , {circumflex over (d)}=d−D.which gives${\overset{.}{\hat{x}}}_{1} = {\frac{1}{L}\left\lbrack {{{- \frac{\left( {1 - D} \right)R}{R + R_{c}}}{\hat{x}}_{2}} - {\frac{\left( {1 - D} \right){RR}_{c}}{R + R_{c}}{\hat{x}}_{1}} - {R_{s}{\hat{x}}_{1}} + {\left( {{\frac{{RR}_{c}}{R + R_{c}}x_{10}} + {\frac{R}{R + R_{c}}x_{20}}} \right)\hat{d}}} \right\rbrack}$${\overset{.}{\hat{x}}}_{2} = {\frac{1}{C\left( {R + R_{c}} \right)}\left\lbrack {{- {\hat{x}}_{2}} + {\left( {1 - D} \right)R{\hat{x}}_{1}} - {{Rx}_{10}\hat{d}}} \right\rbrack}$In matrix form $\begin{bmatrix}{\overset{.}{\hat{x}}}_{1} \\{\overset{.}{\hat{x}}}_{2}\end{bmatrix} = {{\begin{bmatrix}{- \left( {\frac{{RR}_{c}\left( {1 - D} \right)}{L\left( {R + R_{c}} \right)} + \frac{R_{s}}{L}} \right)} & {- \frac{\left( {1 - D} \right)R}{L\left( {R + R_{c}} \right)}} \\\frac{R\left( {1 - D} \right)}{C\left( {R + R_{c}} \right)} & {- \frac{1}{C\left( {R + R_{c}} \right)}}\end{bmatrix}\begin{bmatrix}{\hat{x}}_{1} \\{\hat{x}}_{2}\end{bmatrix}} + {\begin{bmatrix}\frac{{{RR}_{c}x_{10}} + {Rx}_{20}}{L\left( {R + R_{c}} \right)} \\\frac{{Rx}_{10}}{C\left( {R + R_{c}} \right)}\end{bmatrix}{\hat{d}.}}}$Making the following substitutions, which can be derived by letting {dotover (x)}₁=0, {dot over (x)}₂=0, x₁=x₁₀, x₂=x₂₀, R_(c)=0, and R_(s)=0 in(35ab): $x_{10} = \frac{u_{0}}{\left( {1 - D} \right)^{2}R}$ and$x_{20} = \frac{u_{0}}{\left( {1 - D} \right)}$results in $\begin{matrix}{\begin{bmatrix}{\overset{.}{\hat{x}}}_{1} \\{\overset{.}{\hat{x}}}_{2}\end{bmatrix} = {\begin{bmatrix}{- \left( {\frac{{RR}_{c}\left( {1 - D} \right)}{L\left( {R + R_{c}} \right)} + \frac{R_{s}}{L}} \right)} & {- \frac{\left( {1 - D} \right)R}{L\left( {R + R_{c}} \right)}} \\\frac{R\left( {1 - D} \right)}{C\left( {R + R_{c}} \right)} & {- \frac{1}{C\left( {R + R_{c}} \right)}}\end{bmatrix}{\quad{\begin{bmatrix}{\hat{x}}_{1} \\{\hat{x}}_{2}\end{bmatrix} + {\begin{bmatrix}{\frac{u_{0}}{\left( {1 - D} \right)}\left( {\frac{R_{c}}{{L\left( {R + R_{c}} \right)}\left( {1 - D} \right)} + \frac{1}{L}} \right)} \\{- \frac{u_{0}}{{C\left( {R + R_{c}} \right)}\left( {1 - D} \right)^{2}}}\end{bmatrix}\hat{d}}}}}} & (39) \\{\hat{y} = {\left\lbrack {\frac{RRc}{R + R_{c}}\frac{R}{R + R_{c}}} \right\rbrack\hat{x}}} & (40)\end{matrix}$

Now a linear system is provided{circumflex over (x)}=A{circumflex over (x)}+B{circumflex over (d)}ŷ=C{circumflex over (x)}  (41)where A is an n×n matrix, B is an n-column vector, and C is an n-rowvector. To find the transfer function, solve the matrix equationGp(s)=C[sI−A] ⁻¹ B.and obtain${G_{p}(s)} = {{{\frac{1}{\Delta\quad(s)}\left\lbrack {\frac{{RR}_{c}}{R + R_{c}}\frac{R}{R + R_{c}}} \right\rbrack}\begin{bmatrix}{s + \frac{1}{C\left( {R + R_{c}} \right)}} & {- \frac{\left( {1 - D} \right)R}{L\left( {R + R_{c}} \right)}} \\\frac{R\left( {1 - D} \right)}{C\left( {R + R_{c}} \right)} & {s + \left( {\frac{{RR}_{c}\left( {1 - D} \right)}{L\left( {R + R_{c}} \right)} + \frac{R_{s}}{L}} \right)}\end{bmatrix}}{\quad\begin{bmatrix}{\frac{u_{0}}{\left( {1 - D} \right)}\left( {\frac{R_{c}}{{L\left( {R + R_{c}} \right)}\left( {1 - D} \right)} + \frac{1}{L}} \right)} \\{- \frac{u_{0}}{{C\left( {R + R_{c}} \right)}\left( {1 - D} \right)^{2}}}\end{bmatrix}}}$

If we set R_(s)=0 and let powers greater than one of R_(c) equal zero,further evaluation results in $\begin{matrix}{{G_{p}(s)} = {\frac{1}{\Delta\quad(s)}\left\lbrack {\frac{1}{{{LC}\left( {R + R_{c}} \right)}^{2}}{\frac{u_{0}}{\left( {1 - D} \right)^{2}}\left\lbrack {{\left( {R + R_{c}} \right)\left( {L - {{RR}_{c}{C\left( {1 - D} \right)}}} \right)s} + \left( {{R^{2}\left( {1 - D} \right)}^{2} + {2{{RR}_{c}\left( {1 - D} \right)}} + {\left( {1 - D} \right)^{2}R_{c}}} \right)} \right\rbrack}} \right\rbrack}} & (42)\end{matrix}$where Δ(s) is the determinant of [sI−A]⁻¹ which is $\begin{matrix}{{\Delta(s)} = {s^{2} + {\frac{{L\left( {R + R_{c}} \right)} + {{RR}_{c}{C\left( {R + R_{c}} \right)}\left( {1 - D} \right)}}{\left( {R + R_{c}} \right)^{2}{LC}}s} + \frac{{{RR}_{c}\left( {1 - D} \right)} + {R^{2}\left( {1 - D} \right)}^{2}}{\left( {R + R_{c}} \right)^{2}{LC}}}} & (43)\end{matrix}$Taking the term in (42) associated with s, the zero of the linear systemneeds to be in the left-half plane, so this term needs to remainpositive. Solving for R_(c)C we have $\begin{matrix}{{R_{c}C} > \frac{L}{R\left( {1 - D} \right)}} & (44)\end{matrix}$

At this point, the transfer function has been shown to be the linearapproximation of the nonlinear system having a left-half plane zerounder constraint (44). The zeros of the transfer function of the linearapproximation of the nonlinear system at x=0 coincide with theeigenvalues of the linear approximation of the zero dynamics of thenonlinear system at η=0. Therefore, the original nonlinear system (35)has asymptotically stable zero dynamics. Furthermore, the followingproposition is associated with the system (25).

Proposition. Suppose the equilibrium η=0 of the zero dynamics of thesystem is locally asymptotically stable and all the roots of thepolynomial p(s) have negative real part. Then the feedback law$\begin{matrix}{u = {\frac{1}{a\left( {\xi,\eta} \right)}\left( {{- {b\left( {\xi,\eta} \right)}} - {c_{0}z_{1}} - {c_{1}z_{2}} - \cdots - {c_{r - 1}z_{r}}} \right)}} & (45)\end{matrix}$locally asymptotically stabilizes the equilibrium (ξ, η)=(0, 0).

The polynomialp(s)=s ^(r) +c _(r−1) s ^(r−1) + . . . +c ₁ s+c ₀   (46)is the characteristic polynomial of the matrix A associated with theclosed loop system (see equations (25) and (45) and recall that z=ξ){dot over (ξ)}=Aξ+Bv{dot over (η)}=q(ξ,η)   (47)where {dot over (ξ)}=Aξ+Bv are the linear part of the system and {dotover (ξ)}=q(0,η) are the zero dynamics. The matrix A is given by$A = {\begin{bmatrix}0 & 1 & 0 & \cdots & 0 \\0 & 0 & 1 & \cdots & 0 \\\vdots & \vdots & \vdots & \vdots & \vdots \\0 & 0 & 0 & \cdots & 1 \\{- c_{0}} & {- c_{1}} & {- c_{2}} & \cdots & {- c_{r - 1}}\end{bmatrix}.}$and the vector B is given byB=[0, . . . , 0, 1]^(T).

The feedback law in equation (45) can be expressed in the originalcoordinates as $\begin{matrix}{u = {\frac{1}{L_{g}L_{f}^{r - 1}{h(x)}}\left( {{{- L_{f}^{r}}{h(x)}} - {c_{0}{h(x)}} - {c_{1}L_{f}{h(x)}} - \cdots - {c_{r - 1}L_{f}^{r - 1}{h(x)}}} \right)}} & (48)\end{matrix}$

As shown in equation (37), the input d appears after only onedifferentiation so the relative degree is one. This means that thepresent invention is a single order linear system containing only oneroot, thus the present invention can be expressed in the new coordinatesas{dot over (ξ)}=−kξ+v{dot over (η)}=q(ξ,η)y=ξ

The polynomial p(s) is simply p(s)=s+k , with k>0, so that thedenominator is now a real pole in the left half plane.

In accordance with the Proposition, the root of the polynomial p(s) hasa negative real part, and as shown above, the present invention hasasymptotically stable zero dynamics. Therefore, it can be concludedthat, given a control law of the form (48), the original nonlinearsystem (35) is locally asymptotically stable.

The following theorem has been proven.

Theorem 1: For a boost converter with asymptotically stable zerodynamics (using leading-edge modulation),with constraint $\begin{matrix}{{R_{c}C} > \frac{L}{R\left( {1 - D} \right)}} & (49)\end{matrix}$and control law $\begin{matrix}{{d = {\frac{1}{L_{g}L_{f}^{r - 1}y}\left( {{{- L_{f}^{r}}y} - {k\left( {y - y_{0}} \right)}} \right)}};} & (50)\end{matrix}$the nonlinear system{dot over (ξ)}=−kξ+v{dot over (η)}=q(ξ,η),y=ξ  (51)with v=0, is asymptotically stable at each equilibrium point (thecharacteristic polynomial p(s) has a root with negative real part) whichmeans that the original nonlinear system{dot over (x)}=f(x)+g(x)dy=h(x)   (52)is locally asymptotically stable at each equilibrium point (x₁₀, x₂₀) inthe setS={(x ₁ ,x ₂),x εR ^(n) :x ₁≧0,x ₂>0}with 0≦d<1.

Recall that (x₁₀, x₂₀) corresponds to y₀ through equation (36). Theorem1 indicates local asymptotic stability. In practice, the reference inputy₀ is ramped up in a so-called “soft-start” mode of operation. Thistheorem also guarantees local asymptotic stability at each operatingpoint passed through by the system on its way up to the desiredoperating point.

Now referring to FIG. 8, a circuit diagram 800 of a buck-boost converterand a modulator/controller 802 in accordance with the present inventionare shown. The details of buck-boost converters are well known. In thiscase, S2 is implemented with a diode and S1 is implemented with anN-channel MOSFET. FIGS. 9A and 9B are linear circuit diagrams 900 and950 of a buck-boost converter during time DTs and D'Ts. The operation ofthe converter is as follows: u₀ provides power to the circuit during S1conduction time (FIG. 9A) storing energy in inductor L. During this timeS2 is biased off. When S1 turns off, the energy in L causes the voltageacross L to reverse polarity. Since one end is connected to circuitreturn, it remains clamped while the other end forward biases diode S2and clamps to the output. Current continues to flow through L duringthis time (FIG. 9B). When S1 turns back on, the cycle repeats. It shouldbe noted that the output voltage is inverted, i.e., negative. FIG. 10 isa graph of typical waveforms for the buck-boost converter for the twoswitched intervals DTs and D'Ts.

It is again desirable to find the DC transfer function to know how theoutput, y, across the load R is related to the input u₀ at zerofrequency. In steady state, the volt-second integral across L is againequal to zero. Thus,˜₀ ^(T) ^(s) v_(L)dt=0   (53)where Ts is the switching period.

Therefore, the volt-seconds during the on-time must equal thevolt-seconds during the off-time. Using this volt-second balanceconstraint one can derive an equation for volt-seconds during theon-time of S1 (DTs) and another equation for volt-seconds during theoff-time of S1 (D'Ts).

The parasitics are eliminated by setting R_(s)=0 and R_(c)=0.

During time DTs:DT_(s)v_(L)=DT_(s)u₀   (54)During time D'Ts:D'T _(s) v _(L) =−D'Tx _(s) x ₂   (55)

Since by equation (1)DT_(s)v_(L)=D'T_(s)v_(L)The RHS of equation (54) is set equal to the RHS of equation (55) toprovide $\begin{matrix}{\frac{x_{2}}{u_{0}} = {- \frac{D}{D^{\prime}}}} & (56)\end{matrix}$

Equation (56) is the ideal duty ratio equation for the buck-boost cell.If R_(s) and R_(c) are both non-zero then $\begin{matrix}{{\frac{y}{u_{0}} - \frac{x_{1}R_{s}}{D^{\prime}u_{0}}} = {- \frac{D}{D^{\prime}}}} & (57)\end{matrix}$The output y is $\begin{matrix}{y = {{{- D^{\prime}}\frac{{RR}_{c}}{R + R_{c}}x_{1}} + {\frac{R}{R + R_{c}}x_{2}}}} & (58)\end{matrix}$Once again it is seen in equation (57) that parasitic R_(s) should beminimized. For example if R_(s)=0 and R_(c)=0, then equation (57)reduces to the ideal equation (56).

During dTs:${\overset{.}{x}}_{1} = {{\frac{1}{L}u} - {\frac{R_{s}}{L}x_{1}}}$${\overset{.}{x}}_{2} = {{- \frac{1}{C\left( {R + R_{c}} \right)}}x_{2}}$During (1−d)Ts:${\overset{.}{x}}_{1} = {\frac{1}{L}\left( {{{- \left( {R_{s} + \frac{{RR}_{c}}{R + R_{c}}} \right)}x_{1}} + {\frac{R}{R + R_{c}}x_{2}}} \right)}$${\overset{.}{x}}_{2} = {{{- \frac{R}{C\left( {R + R_{c}} \right)}}x_{1}} - {\frac{1}{C\left( {R + R_{c}} \right)}x_{2}}}$Combining, the averaged equations are: $\begin{matrix}{{\overset{.}{x}}_{1} = {{\frac{1}{L}u_{0}d} - {\frac{R_{s}}{L}x_{1}} - {\frac{{RR}_{c}}{L\left( {R + R_{c}} \right)}{x_{1}\left( {1 - d} \right)}} + {\frac{{Rx}_{2}}{L\left( {R + R_{c}} \right)}\left( {1 - d} \right)}}} & \left( {59a} \right) \\\quad & \quad \\{\quad{{{\overset{.}{x}}_{2} = {{{- \frac{R}{\quad{C\left( {R\quad + \quad R_{c}} \right)}}}x_{\quad 1}\left( {1 - d} \right)} - {\frac{1}{\quad{C\left( {R\quad + \quad R_{c}} \right)}}x_{\quad 2}}}}{y = {{\frac{R}{\left( {R + R_{c}} \right)}x_{2}} - {\frac{{RR}_{c}}{\left( {R + R_{c}} \right)}{x_{1}\left( {1 - d} \right)}}}}}} & \left( {{60b},c} \right)\end{matrix}$In standard form: $\begin{matrix}{{{\overset{.}{x}}_{1} = {\frac{{Rx}_{2}}{L\left( {R + R_{c}} \right)} - {\left( {\frac{R_{s}}{L} + \frac{{RR}_{c}}{L\left( {R + R_{c}} \right)}} \right)x_{1}} + {\left( {{\frac{{RR}_{c}}{L\left( {R + R_{c}} \right)}x_{1}} - \frac{{Rx}_{2}}{L\left( {R + R_{c}} \right)} + \frac{u_{0}}{L}} \right)d}}}{{\overset{.}{x}}_{2} = {{{- \frac{1}{C\left( {R + R_{c}} \right)}}x_{2}} - {\frac{R}{C\left( {R + R_{c}} \right)}x_{1}} + {\frac{R}{C\left( {R + R_{c}} \right)}x_{1}d}}}{y = {{{- \frac{{RR}_{c}}{\left( {R + R_{c}} \right)}}x_{1}} + {\frac{R}{\left( {R + R_{c}} \right)}x_{2}}}}} & \left( {{61a},b,c} \right)\end{matrix}$

Here it is assumed that leading-edge modulation is used so that samplingof the output y only takes place during the interval (1−d)T_(s).Therefore, the weighting factor (1−d) in equation (60c) for y has beenremoved in equation (61c) because when the sample is taken the datarepresents both terms. In the present analysis the effects of sampling(complex positive zero pair at one-half the sampling frequency) havebeen ignored.

The output, y, only needs to be differentiated once before the control dappears. Thus, $\begin{matrix}{y = {\frac{R}{R + {Rc}}\left( {x_{2} - {R_{c}x_{1}}} \right)}} & (62) \\{\overset{.}{y} = {\frac{R}{R + {Rc}}\left( {{\overset{.}{x}}_{2} - {R_{c}{\overset{.}{x}}_{1}}} \right)}} & \left( {63a} \right) \\{\overset{.}{y} = {\left( \frac{R}{R + {Rc}} \right)\left\lbrack {{\frac{1}{C\left( {R + R_{c}} \right)}\left( {{- x_{2}} - {Rx}_{1} + {{Rx}_{1}d}} \right)} - {R_{c}\left( {\frac{1}{L}\left( {{\frac{R}{R + R_{c}}x_{2}} - {\left( {R_{s} + \frac{{RR}_{c}}{R + R_{c}}} \right)x_{1}} + {\left( {{\frac{{RR}_{c}}{R + R_{c}}x_{1}} - {\frac{R}{R + R_{c}}x_{2}} + u_{0}} \right)d}} \right)} \right)}} \right\rbrack}} & \left( {63b} \right)\end{matrix}$Substituting for x₂ from (62), setting {dot over (y)} to zero andsolving for d provides, $\begin{matrix}{d = {- \frac{\begin{matrix}{{\left\lbrack {{{RR}_{\quad c}C} + L} \right\rbrack\frac{\left( {R\quad + \quad R_{\quad c}} \right)}{R}y} -} \\{{\left\lbrack {{\left( {R + R_{\quad c}} \right)R_{\quad c}R_{\quad s}C} - {L\left( {R + R_{\quad c}} \right)}} \right\rbrack x_{\quad 1}} + {k\left( {y - y_{\quad 0}} \right)}}\end{matrix}}{{\left( {R + R_{c}} \right)R_{c}{Cy}} + {LRx}_{1} - {\left( {R + R_{c}} \right)R_{c}{Cu}_{0}}}}} & (64)\end{matrix}$where the error term k(y−y₀) has been added. Here y₀ is the desiredoutput corresponding to x₁₀ and x₂₀ through equation (62). The notationhas changed and k=c₀ in equation (22), and the control input is now dinstead of u. Here (x₁₀, x₂₀) is an equilibrium point of our buck-boostconverter. Implementation of the control is the same as shown in FIG. 2.The same definitions are used, so the local linearization will bediscussed.

To obtain the transfer function, a Taylor Series linearization is againused on the nonlinear system (61) to linearize about an operating point,x₁₀, x₂₀, D to provide{circumflex over (x)} ₁ =x ₁ −x ₁₀ , {circumflex over (x)} ₂ =x ₂ −x ₂₀, ŷ=y−y ₀ , {circumflex over (d)}=d ₁ −D.This gives${\overset{.}{\hat{x}}}_{1} = {\frac{1}{L}\left\lbrack {{{- \left( {R_{s} + {\frac{{RR}_{c}}{R + R_{c}}\left( {1 - D} \right)}} \right)}{\hat{x}}_{1}} + {\frac{R}{R + R_{c}}\left( {1 - D} \right){\hat{x}}_{2}} + {\left( {u_{0} + {\frac{{RR}_{c}}{R + R_{c}}x_{10}} - {\frac{R}{R + R_{c}}x_{20}}} \right)\hat{d}}} \right\rbrack}$${\overset{.}{\hat{x}}}_{2} = {\frac{1}{C\left( {R + R_{c}} \right)}\left\lbrack {{- {\hat{x}}_{2}} - {\left( {1 - D} \right)R{\hat{x}}_{1}} + {{Rx}_{10}\hat{d}}} \right\rbrack}$In matrix form $\begin{bmatrix}{\overset{.}{\hat{x}}}_{1} \\\overset{.}{\hat{x}}\end{bmatrix} = {\begin{bmatrix}{- \left( {\frac{{RR}_{c}\left( {1 - D} \right)}{L\left( {R + R_{c}} \right)} + \frac{R_{s}}{L}} \right)} & \frac{R\left( {1 - D} \right)}{L\left( {R + R_{c}} \right)} \\{- \frac{R\left( {1 - D} \right)}{C\left( {R + R_{c}} \right)}} & {- \frac{1}{C\left( {R + R_{c}} \right)}}\end{bmatrix}{\quad{\begin{bmatrix}{\hat{x}}_{1} \\{\hat{x}}_{2}\end{bmatrix} + {\begin{bmatrix}{\frac{{RR}_{c}x_{10}}{L\left( {R + R_{c}} \right)} - \frac{{Rx}_{20}}{L\left( {R + R_{c}} \right)} + \frac{u_{0}}{L}} \\\frac{{Rx}_{10}}{C\left( {R + R_{c}} \right)}\end{bmatrix}{\hat{d}.}}}}}$Making the following substitutions, which can be derived by letting {dotover (x)}₁=0, {dot over (x)}₂=0, x₁=x₁₀, x₂=x₂₀, R_(c)=0, and R_(s)=0 in(61ab): $\begin{matrix}{{x_{10}\quad = \quad\frac{\quad{Du}_{0}}{\quad{\left( {1 - D} \right)^{2}\quad R}}}{and}{x_{20}\quad = {- \frac{\quad{Du}_{0}}{\left( {1 - D} \right)}}}{{{to}\quad{{get}\begin{bmatrix}{\overset{\quad.}{\hat{x}}}_{1} \\{\overset{\quad.}{\hat{x}}}_{2}\end{bmatrix}}} = {\quad{\begin{bmatrix}{- \left( {\frac{\quad{{RR}_{c}\left( {1 - D} \right)}}{L\left( {R + R_{c}} \right)} + \frac{\quad R_{s}}{L}} \right)} & \frac{R\left( {1 - D} \right)}{L\quad\left( {R + R_{c}} \right)} \\\frac{R\quad\left( {1 - D} \right)}{C\left( {R + R_{c}} \right)} & {- \frac{1}{C\quad\left( {R + R_{c}} \right)}}\end{bmatrix}{\quad{\begin{bmatrix}{\hat{x}}_{1} \\{\hat{x}}_{2}\end{bmatrix} + {\begin{bmatrix}{{\frac{{Du}_{0}}{\left( {1 - D} \right)\quad L}\quad\left( {\frac{R_{c}}{\left( {R + R_{c}} \right)\quad\left( {1 - D} \right)}\quad + \frac{R}{\left( {R + R_{c}} \right)}} \right)} + \frac{u_{0}}{L}} \\\frac{u_{0}D}{C\quad\left( {R + R_{c}} \right)\left( {1 - D} \right)^{2}}\end{bmatrix}\hat{d}}}}}}}} & (65) \\{\hat{y} = {\left\lbrack {{- \frac{RRc}{R + R_{c}}}\frac{R}{R + R_{c}}} \right\rbrack\hat{x}}} & (66)\end{matrix}$

Now the linear system{circumflex over (x)}=A{circumflex over (x)}+B{circumflex over (d)}ŷ=C{circumflex over (x)}  (67)is provided where A is an n x n matrix, B is an n-column vector, and Cis an n-row vector. To find the transfer function the matrix equation issolvedG _(p)(s)=C[sI−A] ⁻¹ B.After some algebra, setting R_(s)=0 and letting powers greater than oneof R_(c) equal zero provides $\begin{matrix}{{G_{p}(s)} = {{- \frac{1}{\Delta\quad(s)}}\left\{ {\frac{1}{{{LC}\left( {R + R_{c}} \right)}^{2}}\frac{{Du}_{0}}{\left( {1 - D} \right)^{2}}\left( {{\left\lbrack {{LD} - {{RR}_{c}{C\left( {1 - D} \right)}}} \right\rbrack s} + {\left( {1 - D} \right){R\left\lbrack {{\left( {D^{2} - D + 2} \right)R_{c}} + {\left( {1 - D} \right)R}} \right\rbrack}}} \right)} \right\}}} & (68)\end{matrix}$where Δ(s) is the determinant of [sI−A]⁻¹ which is $\begin{matrix}\begin{matrix}{{\Delta(s)} = {s^{2} + {\left( \frac{L + {{RR}_{c}{C\left( {1 - D} \right)}\left( {R + R_{c}} \right)}}{\left( {R + R_{c}} \right)^{2}{LC}} \right)s} +}} \\{\frac{\left\lbrack {{{RR}_{c}\left( {1 - D} \right)} + {R^{2}\left( {1 - D} \right)}^{2}} \right\rbrack}{\left( {R + R_{c}} \right)^{2}{LC}}}\end{matrix} & (69)\end{matrix}$

Taking the term in (68) associated with s, the zero of the linearapproximation of the system should be in the left-half plane, so thisterm needs to remain positive. Solving for R_(c)C results in$\begin{matrix}{{R_{c}C} > {\frac{LD}{R\left( {1 - D} \right)}.}} & (70)\end{matrix}$

At this point, it has been shown that the transfer function of thelinear approximation of the nonlinear system has a left-half plane zerounder constraint (70). As before, it is known that the zeros of thetransfer function of the linear approximation of the nonlinear system atx=0 coincide with the eigenvalues of the linear approximation of thezero dynamics of the nonlinear system at η=0. Therefore, the originalnonlinear system (61) has asymptotically stable zero dynamics.

The Proposition is again used, with p(s) as above in equation (46) andthe closed loop system as in equation (47). As shown in equation (63),the input d appears after only one differentiation so the relativedegree is again one. This means that the present invention is a singleorder linear system containing only one root, thus the present inventioncan be expressed in the new coordinates as{dot over (ξ)}=−kξ+v{dot over (η)}=q(ξ,η)y=ξ.The polynomial p(s) is simply p(s)=s+k , with k>0, so that thedenominator is now a real pole in the left half plane.

In accordance with the Proposition, the root of the polynomial p(s) hasa negative real part, and as shown above, the present invention hasasymptotically stable zero dynamics. Therefore, given a control law ofthe form (48), it can be concluded that the original nonlinear system(61) is locally asymptotically stable.

The following theorem has been proven.

Theorem 2: For a buck-boost converter with asymptotically stable zerodynamics (using leading-edge modulation), with constraint$\begin{matrix}{{R_{c}C} > \frac{LD}{R\left( {1 - D} \right)}} & (71)\end{matrix}$and control law $\begin{matrix}{{d = {\frac{1}{L_{g}L_{f}^{r - 1}y}\left( {{{- L_{f}^{r}}y} - {k\left( {y - y_{0}} \right)}} \right)}};} & (72)\end{matrix}$the nonlinear system{dot over (ξ)}=−kξ+v{dot over (η)}=q(ξ,η),y=ξ  (73)with v=0, is asymptotically stable at each equilibrium point (thecharacteristic polynomial p(s) has a root with negative real part) whichmeans that the original nonlinear system{dot over (x)}=f(x)+g(x)dy=h(x)   (74)is locally asymptotically stable at each equilibrium point (x₁₀, x₂₀) inthe setS={(x₁ ,x ₂),x εR ^(n) :x ₁≧0,x ₂≦0}with 0≦d<1.

Recall that (x₁₀, x₂₀) corresponds to y₀ through equation (62).

Theorem 2 indicates local asymptotic stability. In practice, thereference input y₀ is ramped up in a so-called “soft-start” mode ofoperation. This theorem also guarantees local asymptotic stability ateach operating point passed through by the system on its way up to thedesired operating point.

Now referring to FIGS. 11A, 11B and 12, the boost converter shown inFIG. 5 is simulated using Simulink under a switching regime usinginput-output feedback linearization and LEM. Two dynamic cases aresimulated. In the first case, the operating point voltage, y₀, wasstepped from 28V to 32V to 28V with k=0.3. The output response appearsin FIG. 11A. A closeup of the output voltage transition from 28V to 32Vis shown in FIG. 11B. The high switching ripple is due to the value ofR_(c).

In FIG. 11A above, when t=5 mS, the switching turns completely off andthe output capacitor is discharged into load R until the output voltagey reaches the desired voltage of 28V which causes the switching to startup again to maintain the regulated output. The close up view in FIG. 11Babove shows the RHP zero effect in that the pulse width is commanded tomaximum and yet the output voltage drops (see bottom of the ripplewaveform). The top of the waveform does not drop and this is the voltagesampled during the interval 1−d giving us the desired effect of movingthe RHP zero into the LHP.

In the second case, with y₀=28V, the output load R was allowed to changefrom 14 Ω to 7 Ω to 14 Ω, which results in a load step of 2A to 4A to2A. The transient response of y resulting from these changes is shown inFIG. 12. In both cases, k is kept at the same value. In addition, slopecompensation with a positive slope was added to duty ratio d. Note thatslope compensation can be added.

Referring now to FIGS. 13, 14, 15, the buck-boost converter shown inFIG. 8 is simulated using Simulink under a switching regime usinginput-output feedback linearization and LEM. Two dynamic cases aresimulated. In the first case, the operating point voltage, y₀, wasstepped from −18V to −32V with k=0.3. The output response appears inFIG. 13. In the second case, with y₀=−18V, the output load R was allowedto change from 6 Ω to 3 Ω to 6 Ω, which results in a load step of 3A to6A to 3A. The transient response of y resulting from these changes isshown in FIG. 14. In both cases, k is kept at the same value. Inaddition, slope compensation with a positive slope was added to dutyratio d.

The transition from −32V to −18V is not shown in FIG. 13 because theeffect is similar to FIG. 11A in that switching turns completely off,and the output capacitor is discharged into load R until the outputvoltage y reaches the desired voltage of −18V, causing the switching tostart up again to maintain the regulated output. The transition duringthe load step is clean and smooth as seen in FIG. 14.

Input-output feedback linearization will now be applied to the nonlinearstate space averaged models of both the boost and buck-boost converters.The analysis will be restricted to the switched regime using a new modelcalled the Discrete Average Model using a switched Spice circuit insteadof averaged equations. This new model used the averaged states but theoutput is sampled at the switching frequency by using leading-edgemodulation instead of trailing-edge modulation. In fact, trailing-edgemodulation could not be used because of the unstable zero dynamicsassociated with it during the continuous conduction mode of operation.Using leading-edge modulation, the unstable zero dynamics are moved tostable zero dynamics and establish a new constraint. For DC-DCconversion, equivalent equations (38) and (64) are derived for the dutyratio d that involve only the linear parameter k and any state spaceoperating point (x₁₀, x₂₀). With this d, the equilibrium point (x₁₀,x₂₀) is asymptotically stable as shown in Theorems 1 and 2.

Now referring to FIG. 15, a Spice analysis of the buck-boost converter1500 using input-output linearization with leading-edge modulation willnow be described. The two dynamic cases previously done with Simulinkwere again simulated under the same conditions. The first case is for astep in y₀ from −18V to −32V. Only this direction of step is shown sincethe step from −32V to −18V results in the trivial result of the PWMturning completely off and waiting for the output capacitor to dischargeup to −18V.

The circuit description for y₀ step will now be described. Power switchM1 is a P-channel enhancement mode MOSFET. The model chosen is anIRFP9240. However, in order to closely model the ideal switches used inSimulink, the R_(ds(on)) of M1 was changed to one-tenth of a milliOhm(10⁻⁴ Ohms). This entailed changing both ‘rd’ and ‘rs’ in the model. Thefunction B9 measures x₁ through L₁. It simply takes the voltage at nodeR₁ and divides it by 0.02. The output voltage y is taken directly fromthe node labeled y in FIG. 15. The diode D1 is a generic fast recoverydiode. The transit time was decreased to simulate a hyperfast recoverydiode. This results in lower switching spikes on the output ripple. Forexample, a PWM modulator is included in the control circuit of FIG. 16,which is included only as an example and is not intended to limit theinvention in any way. Those skilled in the art will recognize that FIGS.16 and 17 represent one specific example and that the specificapplication and modeling software will dictate the exact schematicdiagram necessary to implement the present invention.

The sawtooth ramp with a negative slope (LEM) is generated by element V3and is connected to the inverting input of the comparator. Thecomparator model is a MAX907CPA high slew rate device with a lowpropagation delay. The non-inverting input is connected to the output ofa limiter which actually sets the maximum duty ratio. In this example,the maximum duty ratio is set to 0.8. The element B7 actually calculatesthe control voltage, which is compared against the sawtooth ramp. Thecalculations are based on the elements shown in FIG. 17, which isincluded only as an example and is not intended to limit the inventionin any way. Element V6 provides the step in the reference y₀.

The PWM modulator output steering logic was taken from an applicationnote for the SG1526 PWM IC originally manufactured by Silicon General.The drive transistors were then added to give sufficient currentcapability to drive the high capacitive load of the MOSFET gate. Lastly,the two channel outputs were diode-or'ed to become a single channel. Theresistors R4 and R5 with ramp generator V12 provide slope compensation.The value of m_(c) here is 1.5. The slope of the ramp generated by V12is positive which offsets the ramp slope generated by ramp generator V3.The control voltage labeled d in FIG. 16 represents the duty ratio. Itis converted to PWM by comparison with the sawtooth ramp.

Simulation results for y₀ will now be discussed. In the first case, theoperating point voltage, y₀, is stepped from −18V to −32V with k=0.3.The output response is shown in FIG. 18 wherein y is labeled V(out) andy₀ is labeled V(ref). As seen by comparison of FIG. 18 with the Simulinksimulation of FIG. 13, the similarity in the response is remarkable.This result justifies the accuracy of the previous models and the use ofSimulink in modeling and simulating such switching converters.

A circuit description for R step will now be discussed. The circuitdescription given above applies here as well with the exception thathere the load R was changed whereas y₀ was changed before. The load R ischanged from 6 Ω to 3 Ω to 6 Ω. This is accomplished with switch S₁ asshown in FIG. 19. Switch S₁ is pulsed on (closed) for 1 mS by generatorV12 at node ‘pulse.’ The PWM modulator and control are the same as shownin FIG. 16 except that reference voltage V6 is now fixed at −18V. Theelements in FIG. 17 remain the same.

The simulation results for R step will now be discussed. In the secondcase, the load, R, is stepped from 6 Ω to 3 Ω to 6 Ω with k=0.3. Theoutput response is shown in FIG. 20 wherein y is labeled V(out), y₀ islabeled V(ref) and R is labeled V(r). Comparing FIG. 20 with theSimulink simulation of FIG. 14, noticeably similar results can be seen.

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Although preferred embodiments of the present invention have beendescribed in detail, it will be understood by those skilled in the artthat various modifications can be made therein without departing fromthe spirit and scope of the invention as set forth in the appendedclaims.

1. A method for controlling a non-linear system comprising the steps of:receiving a first voltage, a second voltage, a first current and asecond current; creating a third voltage representing a differencebetween the first voltage and the second voltage; adjusting the thirdvoltage by a proportional gain; creating a control signal based on thesecond voltage, the adjusted third voltage, the first current and thesecond current; and controlling the non-linear system using the controlsignal.
 2. The method as recited in claim 1, wherein the first voltageis an output voltage from the converter, the second voltage is areference voltage, the first current is an inductor current from theconverter and the second current is a reference current.
 3. The methodas recited in claim 1, wherein the non-linear system is a boostconverter or a buck-boost converter.
 4. The method as recited in claim1, wherein the non-linear system is a boost converter and a duty cyclefor the control signal is defined as $d = {- {\frac{\begin{matrix}{{\left\lbrack {{{CR}_{s}{R_{c}\left( {R + R_{c}} \right)}} + {L\left( {R + R_{c}} \right)}} \right\rbrack x_{1}} + \left\lbrack {{\left( {R + R_{c}} \right)R_{c}C} + L} \right\rbrack} \\{{\frac{\left( {R + R_{c}} \right)}{R}y} - {\left( {R + R_{c}} \right)R_{c}{Cu}_{0}} - {k\left( {y - y_{0}} \right)}}\end{matrix}}{{LRx}_{1} + {\left\lbrack {\left( {R + R_{c}} \right)R_{c}C} \right\rbrack\frac{\left( {R + R_{c}} \right)}{R}y}}.}}$5. The method as recited in claim 1, wherein the non-linear system is abuck-boost converter and a duty cycle for the control signal is definedas $d = {- {\frac{\begin{matrix}{\left\lbrack {{{RR}_{c}C} + L} \right\rbrack\frac{\left( {R + R_{c}} \right)}{R}} \\{y - {\left\lbrack {{\left( {R + R_{c}} \right)R_{c}R_{s}C} - {L\left( {R + R_{c}} \right)}} \right\rbrack x_{1}} + {k\left( {y - y_{0}} \right)}}\end{matrix}}{{\left( {R + R_{c}} \right)R_{c}C\quad y} + {LRx}_{1} - {\left( {R + R_{c}} \right)R_{c}{Cu}_{0}}}.}}$6. The method as recited in claim 1, wherein the control signal providesleading-edge modulation with input-output linearization.
 7. The methodas recited in claim 1, wherein the control of the non-linear systemcannot be unstable.
 8. The method as recited in claim 1, wherein thecontrol signal is created using a first order system.
 9. The method asrecited in claim 1, wherein the control signal is independent of astabilizing gain, a desired output voltage or a desired outputtrajectory.
 10. A converter controlled in accordance with the method ofclaim
 1. 11. A computer program embodied within a digital signalprocessor for controlling a non-linear system, the computer programcomprising: a code segment for receiving a first voltage, a secondvoltage, a first current and a second current; a code segment forcreating a third voltage representing a difference between the firstvoltage and the second voltage; a code segment for adjusting the thirdvoltage by a proportional gain; a code segment for creating a controlsignal based on the second voltage, the adjusted third voltage, thefirst current and the second current; and a code segment for controllingthe non-linear system using the control signal.
 12. The computer programas recited in claim 11, wherein the first voltage is an output voltagefrom the converter, the second voltage is a reference voltage, the firstcurrent is an inductor current from the converter and the second currentis a reference current.
 13. The computer program as recited in claim 11,wherein the non-linear system is a boost converter or a buck-boostconverter.
 14. The computer program as recited in claim 11, wherein thenon-linear system is a boost converter and a duty cycle for the controlsignal is defined as $d = {- {\frac{\begin{matrix}{{\left\lbrack {{{CR}_{s}{R_{c}\left( {R + R_{c}} \right)}} + {L\left( {R + R_{c}} \right)}} \right\rbrack x_{1}} + \left\lbrack {{\left( {R + R_{c}} \right)R_{c}C} + L} \right\rbrack} \\{{\frac{\left( {R + R_{c}} \right)}{R}y} - {\left( {R + R_{c}} \right)R_{c}{Cu}_{0}} - {k\left( {y - y_{0}} \right)}}\end{matrix}}{{LRx}_{1} + {\left\lbrack {\left( {R + R_{c}} \right)R_{c}C} \right\rbrack\frac{\left( {R + R_{c}} \right)}{R}y}}.}}$15. The computer program as recited in claim 11, wherein the non-linearsystem is a buck-boost converter and a duty cycle for the control signalis defined as $d = {- {\frac{\begin{matrix}{\left\lbrack {{{RR}_{c}C} + L} \right\rbrack\frac{\left( {R + R_{c}} \right)}{R}} \\{y - {\left\lbrack {{\left( {R + R_{c}} \right)R_{c}R_{s}C} - {L\left( {R + R_{c}} \right)}} \right\rbrack x_{1}} + {k\left( {y - y_{0}} \right)}}\end{matrix}}{{\left( {R + R_{c}} \right)R_{c}C\quad y} + {LRx}_{1} - {\left( {R + R_{c}} \right)R_{c}{Cu}_{0}}}.}}$16. An apparatus comprising one or more electrical circuits that providea control signal to a boost converter such that a duty cycle of thecontrol signal is defined as $d = {- {\frac{\begin{matrix}{{\left\lbrack {{{CR}_{s}{R_{c}\left( {R + R_{c}} \right)}} + {L\left( {R + R_{c}} \right)}} \right\rbrack x_{1}} + \left\lbrack {{\left( {R + R_{c}} \right)R_{c}C} + L} \right\rbrack} \\{{\frac{\left( {R + R_{c}} \right)}{R}y} - {\left( {R + R_{c}} \right)R_{c}{Cu}_{0}} - {k\left( {y - y_{0}} \right)}}\end{matrix}}{{LRx}_{1} + {\left\lbrack {\left( {R + R_{c}} \right)R_{c}C} \right\rbrack\frac{\left( {R + R_{c}} \right)}{R}y}}.}}$17. An apparatus comprising one or more electrical circuits that providea control signal to a buck-boost converter such that a duty cycle of thecontrol signal is defined as $d = {- {\frac{\begin{matrix}{\left\lbrack {{{RR}_{c}C} + L} \right\rbrack\frac{\left( {R + R_{c}} \right)}{R}} \\{y - {\left\lbrack {{\left( {R + R_{c}} \right)R_{c}R_{s}C} - {L\left( {R + R_{c}} \right)}} \right\rbrack x_{1}} + {k\left( {y - y_{0}} \right)}}\end{matrix}}{{\left( {R + R_{c}} \right)R_{c}C\quad y} + {LRx}_{1} - {\left( {R + R_{c}} \right)R_{c}{Cu}_{0}}}.}}$18. An apparatus comprising: a first connection to receive a firstvoltage; a second connection to receive a second voltage; a thirdconnection to receive a first current; a fourth connection to receive asecond current; a fifth connection to output a control signal; a summingcircuit connected to the first connection and the second connection tocreate a third voltage representing a difference between the firstvoltage and the second voltage; a gain circuit connected to the summingcircuit to adjust the third voltage by a proportional gain; and amodulating circuit connected to the gain circuit, the second connection,the third connection, the fourth connection and the fifth connection,the modulation circuit creating a control signal based on the secondvoltage, the adjusted third voltage, the first current and the secondcurrent.
 19. The apparatus as recited in claim 18, wherein the firstvoltage is an output voltage from the converter, the second voltage is areference voltage, the first current is an inductor current from theconverter and the second current is a reference current.
 20. Theapparatus as recited in claim 18, wherein the control signal controls aboost converter or a buck-boost converter.
 21. The apparatus as recitedin claim 18, wherein the control signal controls a boost converter andhas a duty cycle defined as $d = {- {\frac{\begin{matrix}{{\left\lbrack {{{CR}_{s}{R_{c}\left( {R + R_{c}} \right)}} + {L\left( {R + R_{c}} \right)}} \right\rbrack x_{1}} + \left\lbrack {{\left( {R + R_{c}} \right)R_{c}C} + L} \right\rbrack} \\{{\frac{\left( {R + R_{c}} \right)}{R}y} - {\left( {R + R_{c}} \right)R_{c}{Cu}_{0}} - {k\left( {y - y_{0}} \right)}}\end{matrix}}{{LRx}_{1} + {\left\lbrack {\left( {R + R_{c}} \right)R_{c}C} \right\rbrack\frac{\left( {R + R_{c}} \right)}{R}y}}.}}$22. The apparatus as recited in claim 18, wherein the control signalcontrols a buck-boost converter and has a duty cycle defined as$d = {- {\frac{\begin{matrix}{{\left\lbrack {{{RR}_{c}C} + L} \right\rbrack\frac{\left( {R + R_{c}} \right)}{R}y} -} \\{{\left\lbrack {{\left( {R + R_{c}} \right)R_{c}R_{s}C} - {L\left( {R + R_{c}} \right)}} \right\rbrack x_{1}} + {k\left( {y - y_{0}} \right)}}\end{matrix}}{{\left( {R + R_{c}} \right)R_{c}{Cy}} + {LRx}_{1} - {\left( {R + R_{c}} \right)R_{c}{Cu}_{0}}}.}}$23. A system comprising: a converter having a first voltage and a firstcurrent; a second voltage source; a second current source; and a PWMmodulator/controller comprising: a summing circuit connected to theconverter and the second voltage source to create a third voltagerepresenting a difference between the first voltage and the secondvoltage, a gain circuit connected to the summing circuit to adjust thethird voltage by a proportional gain, and a modulating circuit connectedto the gain circuit, the converter, the second voltage second and thesecond current source to create a control signal based on the secondvoltage, the adjusted third voltage, the first current and the secondcurrent, and to the converter to control the converter with the controlsignal.
 24. The system as recited in claim 23, wherein the secondvoltage source and second current source are integrated into the PWMmodulator/controller.
 25. The system as recited in claim 23, wherein thePWM modulator/controller is implemented using a digital signal processoror conventional electrical circuitry.
 26. The system as recited in claim23, wherein the first voltage is an output voltage from the converter,the second voltage is a reference voltage, the first current is aninductor current from the converter and the second current is areference current.
 27. The system as recited in claim 23, wherein thecontrol signal controls a boost converter or a buck-boost converter. 28.The system as recited in claim 23, wherein the control signal controls aboost converter and has a duty cycle defined as$d = {- {\frac{\begin{matrix}{{\left\lbrack {{{CR}_{s}{R_{c}\left( {R + R_{c}} \right)}} + {L\left( {R + R_{c}} \right)}} \right\rbrack x_{1}} + \left\lbrack {{\left( {R + R_{c}} \right)R_{c}C} + L} \right\rbrack} \\{{\frac{\left( {R + R_{c}} \right)}{R}y} - {\left( {R + R_{c}} \right)R_{c}{Cu}_{0}} - {k\left( {y - y_{0}} \right)}}\end{matrix}}{{LRx}_{1} + {\left\lbrack {\left( {R + R_{c}} \right)R_{c}C} \right\rbrack\frac{\left( {R + R_{c}} \right)}{R}y}}.}}$29. The system as recited in claim 23, wherein the control signalcontrols a buck-boost converter and has a duty cycle defined as$d = {- {\frac{\begin{matrix}{\left\lbrack {{{RR}_{c}C} + L} \right\rbrack\frac{\left( {R + R_{c}} \right)}{R}} \\{y - {\left\lbrack {{\left( {R + R_{c}} \right)R_{c}R_{s}C} - {L\left( {R + R_{c}} \right)}} \right\rbrack x_{1}} + {k\left( {y - y_{0}} \right)}}\end{matrix}}{{\left( {R + R_{c}} \right)R_{c}{Cy}} + {LRx}_{1} - {\left( {R + R_{c}} \right)R_{c}{Cu}_{0}}}.}}$30. A kit for a PWM modulated converter comprising; a digital signalprocessor comprising: a first connection to receive a first voltage, asecond connection to receive a second voltage, a third connection toreceive a first current, a fourth connection to receive a secondcurrent, a fifth connection to output a control signal, a summingcircuit connected to the first connection and the second connection tocreate a third voltage representing a difference between the firstvoltage and the second voltage, a gain circuit connected to the summingcircuit to adjust the third voltage by a proportional gain, and amodulating circuit connected to the gain circuit, the second connection,the third connection, the fourth connection and the fifth connection,the modulation circuit creating a control signal based on the secondvoltage, the adjusted third voltage, the first current and the secondcurrent; and a computer program embodied on a computer readable mediumfor programming the digital signal processor to control the PWMmodulated converter.
 31. The kit as recited in claim 30, wherein thecomputer program further comprises one or more design tools.